Hilbert Space of the Bicomplex Quantum Harmonic Oscillator

Bicomplex numbers are pairs of complex numbers with a multiplication law that makes them a commutative ring. The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers. Starting with the commutator of the bicomplex position and momentum operators, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. Coordinate-basis eigenfunctions of the Hamiltonian are then obtained in terms of hyperbolic Hermite polynomials, and some of them are graphically illustrated. These eigenfunctions form a basis of an infinite-dimensional module over bicomplex numbers, and this module can be given the structure of a bicomplex Hilbert space.